Find The Root: F(x) = 60x^2 - 57x - 18

by Alex Johnson 39 views

Let's dive into finding the actual root of the quadratic function f(x) = 60x^2 - 57x - 18. We'll explore how to apply the Rational Root Theorem and then test the potential roots to see which one satisfies the equation. This involves a blend of algebraic understanding and methodical testing, ensuring we arrive at the correct solution. So, let's get started and unravel this mathematical puzzle step by step.

Understanding the Rational Root Theorem

To solve this, we need to understand the Rational Root Theorem. This theorem is a gem in algebra that helps us identify potential rational roots of a polynomial equation. In simpler terms, it narrows down the list of possible solutions, making our search much more manageable. The theorem states that if a polynomial has integer coefficients, then every rational root of the polynomial has the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.

In our case, the polynomial is f(x) = 60x^2 - 57x - 18. Here:

  • The constant term is -18.
  • The leading coefficient is 60.

So, to apply the Rational Root Theorem:

  1. List the factors of the constant term (-18): ±1, ±2, ±3, ±6, ±9, ±18.
  2. List the factors of the leading coefficient (60): ±1, ±2, ±3, ±4, ±5, ±6, ±10, ±12, ±15, ±20, ±30, ±60.
  3. Form all possible fractions p/q, where p is a factor of -18 and q is a factor of 60. This will give us a list of potential rational roots.

This process might seem a bit lengthy, but it's a systematic way to find potential roots. Now, let's look at the potential roots provided and see how they fit into this framework.

Potential Roots: A Closer Look

The question gives us four potential roots:

  • -6/5
  • -1/4
  • 3
  • 6

These values have been derived using the Rational Root Theorem, so they are indeed potential candidates. Our next step is to test each of these to see which one, if any, is an actual root of the equation f(x) = 60x^2 - 57x - 18. Remember, a root of a function is a value that makes the function equal to zero. So, we will substitute each potential root into the equation and check if the result is zero.

Testing -6/5

Let's start with -6/5. We substitute x = -6/5 into the equation:

f(-6/5) = 60(-6/5)^2 - 57(-6/5) - 18

Now, we need to carefully calculate this expression. First, square -6/5:

(-6/5)^2 = 36/25

Next, multiply by 60:

60 * (36/25) = (60 * 36) / 25 = 2160 / 25 = 86.4

Then, multiply -57 by -6/5:

-57 * (-6/5) = 342 / 5 = 68.4

Now, substitute these back into the equation:

f(-6/5) = 86.4 + 68.4 - 18

Finally, calculate the result:

f(-6/5) = 136.8 ≠ 0

Since f(-6/5) is not equal to zero, -6/5 is not a root of the equation.

Testing -1/4

Next, we test x = -1/4:

f(-1/4) = 60(-1/4)^2 - 57(-1/4) - 18

First, square -1/4:

(-1/4)^2 = 1/16

Multiply by 60:

60 * (1/16) = 60/16 = 15/4 = 3.75

Multiply -57 by -1/4:

-57 * (-1/4) = 57/4 = 14.25

Substitute these back into the equation:

f(-1/4) = 3.75 + 14.25 - 18

Calculate the result:

f(-1/4) = 18 - 18 = 0

Since f(-1/4) = 0, -1/4 is a root of the equation!

Testing 3 and 6 (Optional for Demonstration)

To further illustrate the process, let's quickly test the remaining options, although we've already found a root. This will solidify our understanding and demonstrate why the other options are not roots.

Testing 3

Substitute x = 3 into the equation:

f(3) = 60(3)^2 - 57(3) - 18

Calculate:

f(3) = 60 * 9 - 171 - 18 f(3) = 540 - 171 - 18 f(3) = 351 ≠ 0

So, 3 is not a root.

Testing 6

Substitute x = 6 into the equation:

f(6) = 60(6)^2 - 57(6) - 18

Calculate:

f(6) = 60 * 36 - 342 - 18 f(6) = 2160 - 342 - 18 f(6) = 1800 ≠ 0

Thus, 6 is not a root either.

Conclusion: The Actual Root

After systematically testing each potential root using the Rational Root Theorem and substitution, we found that -1/4 is the actual root of the quadratic function f(x) = 60x^2 - 57x - 18. This process demonstrates the power of the Rational Root Theorem in narrowing down potential solutions and the importance of methodical testing to confirm the actual roots.

Understanding and applying theorems like the Rational Root Theorem is fundamental in mathematics. It not only helps in solving equations but also enhances problem-solving skills. By breaking down the problem into manageable steps and meticulously working through each one, we can confidently arrive at the correct answer. If you're interested in learning more about mathematical theorems and their applications, you might find valuable resources on websites like Khan Academy. Happy problem-solving!